Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{t^2 - 25}{t - 5}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = t$ $ b = \sqrt{25} = -5$ So we can rewrite the expression as: $z = \dfrac{({t} {-5})({t} + {5})} {t - 5} $ We can divide the numerator and denominator by $(t - 5)$ on condition that $t \neq 5$ Therefore $z = t + 5; t \neq 5$